Week 5

Empirical validation and verification of error bounds and edge detection methods; Two-dimensional Fourier sums.

Empirical Evaluation of Error Bounds

Last week, you worked out analytical error bounds for the Fourier partial sum reconstruction

as well as the edge-augmented Fourier sum (assuming true jump locations and heights)

Verify these bounds with numerical simulations in Matlab.

Pen and Paper Exercises

Error Bound for Edge-Augmented Fourier Reconstruction with Estimated Edges

Can you extend your analytical results from last week to the case where you use approximate or estimated edge information?

Empirical Evaluation of Edge Detection/Estimation Schemes

• Evaluate the accuracy, efficiency and robustness of the concentration kernel based and Prony-based edge detection methods.
  • Accuracy: Generate plots of the maximum (absolute) error in the jump locations and jump heights for different values of N. Tabulate your results.
  • Computational Cost: Plot the execution time (averaged over several trials) as a function of N. Does this indicate linear/quadratic/other computational complexity?
  • (Optional) Robustness to Measurement Errors: Investigate what happens when the Fourier coefficients are corrupted by additive measurement noise. You can generate measurement noise in Matlab as follows: sig2*randn(size(coefficients)) Here, sig2 is the variance of noise and coefficients is a vector of Fourier coefficients. Plot/tabulate the error in jump locations and jump heights as a function of the added noise variance (Note: when performing experiments with random noise, you may want to report the average error over several trials).
• Repeat the above experiments for your edge-augmented Fourier reconstructions; i.e., when you use the estimated jump information to improve the accuracy of your Fourier reconstruction. As before, evaluate the method based on:
  • Accuracy: Generate plots of the reconstruction and reconstruction error for different values of N. Tabulate the (2-norm and infinity-norm) reconstruction error as a function of N.
  • Computational Cost: Plot the execution time (averaged over several trials) as a function of N. Does this indicate linear/quadratic/other computational complexity?
  • (Optional) Robustness to Measurement Errors: Investigate what happens when the Fourier coefficients are corrupted by additive measurement noise. Plot the (2-norm and infinity norm) reconstruction error as a function of the added noise variance.
• What happens when the number of jumps in the function is not exactly known for the Prony-based method?

Note: Try and write Matlab code (that you can eventually make public on the web) that is readable, well-documented and can recreate your figures/tables/other technical results.

Documents to Prepare

• (due end of day Thurs., 6/23) Prepare a 15 minute Beamer/LaTeX presentation summarizing your research findings from the week. You will be presenting (as a group) to your fellow REU students/groups during the Friday meeting/discussion at the Holmes Hall seminar room. Now that you have explored all facets of the 1D problem, it would be a good time to start generating and refining a complete set of slides - which you would be comfortable presenting at a research seminar to a new audience.
• (due end of day Thurs., 6/23) Continue preparing your technical report in LaTeX detailing your research findings to date. Structure this report as if it were a technical paper - it should contain (i) an abstract (ii) introduction (iii) background material (summarizing the main results/theorems) on Fourier series and the Gibbs phenomenon (iv) theoretical development of edge detection and edge-augmented Fourier reconstructions (v) numerical results. By now, you should have complete 1D theoretical and numerical results.